# Time Dilation Formula

## Time Dilation Formula

The difference in time between two events measured by moving in relation to one another or by gravitational mass or masses at various locations is known as time dilation, according to the theory of relativity.

Think about a clock that has two observers. While one observer is moving at the speed of light, the other is stationary. Time dilation is the existence of a time difference between the two clocks.

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### What is Time Dilation?

Time dilation is a phenomenon in which two bodies that are moving in relation to one another—or even just having different gravitational fields from one another—experience time moving at different rates.

It alludes to a special circumstance in which time can advance at various rates depending on the reference frame. It also depends on how quickly different reference frames move relative to one another. The measurement of elapsed time using two clocks is known as time dilation in layman’s terms. Additionally, two reference frames are the appropriate time (one-position time) and observer time (two-position time). Furthermore, because they are connected, we can calculate the time dilation of one by knowing the speed and velocity of the others.

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### Formula for Time Dilation

The Time Dilation Formula is given by,

T =T0 /√1−(v2/c2)

where,

T is the time observed

T0 is the time observed at rest v is the velocity of the object

c is the velocity of light in a vacuum (3 × 108 m/s2)

### Derivation of Time Dilation

We can link the distances into each other and then calculate each distance in terms of the pulse’s time of travel in the relevant reference frame in order to quantitatively compare the time measurements in the two inertial frames. After that, the resulting equation can be solved for T using T0.

The hypotenuse of a right triangle is made up of the lengths D and L. According to the Pythagorean theorem, s2 = D2 + L2.

The light pulse and the spacecraft have travelled 2s and 2L, respectively, in time in the observer’s frame who is on earth. The length D is the distance covered by the light pulse in time T0 in the astronaut’s frame. As a result, we have three equations to consider:

2L = vT, 2D = cT0, and 2s = cT

We used Einstein’s second postulate by assuming the speed of light to be c in both inertial frames. Now, we can incorporate these findings into the Pythagorean theorem’s initial formulation:

s2 = D2+ L2

(c × T/2)2  = (c × T0/2)2 + (v × T/2)2

Then we rearrange to obtain

(c × T)2 – (v × T)2 = (c × T0)2

Finally, solving for T in terms of T0 gives us

T =T0 /√1−(v/c)2

This is equivalent to

T = γT0,

where  γ is the relativistic factor (often called the Lorentz factor) given by

γ =1/√1−(v2/c2)

and v and c are the speeds of the moving observer and light, respectively.

### Sample Problems

Problem 1: Determine the relativistic time, if T0 is 7 years and the velocity of the object is 0.55c.

Solution:

Given:

T0 = 7 years

v = 0.55c

The Formula for time dilation is given by,

T =T0 /√1−(v2/c2)

T = 7/√1-(0.55)2(32 x 1016)/32 x 1016

T = 7/√1- (0.55)2

T=7/0.8351

T = 8.38 years